In this Master’s Final Project, we develop a survey of determinantal point processes, which form a family of simple point processes whose characteristic property is that their joint intensities $\rho_k(x_1,\ldots,x_k)=\det(K(x_i,x_j))_{1\le i,j\le k}\ \ \forall k\ge1\text{ and }x_1,\ldots,x_k\in E$ are given by the determinants of a kernel $K$ in the appropriate space $E$. Intuitively, these processes describe and allow us to model the behaviour of objects or points that verify certain repulsion interactions between them, determined by their respective kernels. After introducing the necessary preliminaries and using them to define this type of processes, then we proceed to review some of their most remarkable properties. Among them, we especially focus on their existence and uniqueness results, as well as on other selected properties that describe probabilistic behaviours of these processes and justify their interest. Finally, we delve into some specific examples of determinantal point processes, with the aim of showing that they appear naturally in various fields, such as in the study of certain random matrices, whose eigenvalues are examples and give rise to processes of the family. In addition, examples of these processes have also recently appeared in the development of machine learning solutions, something that we also explore in the last section of the project.

Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2026. Director: Joaquim Ortega Cerdà